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In this paper we examine the effectiveness of reducing the second order radial equation, of the hyperspherical coordinate solution to the two-electron Schrodinger equation, into a set of coupled first order linear equations as suggested by Klar. All results have been obtained in a completely nonadiabatic formalism thereby ensuring accuracy. We arrive at the conclusion that our application of the reduction process is in some way inconsistent and suggest a possible resolution to this anomaly.

One of the more recently established methods of analysis of di erentials involves the invariance properties of the equations and the relationship of this with the underlying conservation laws which may be physical. In a variational system, conservation laws are constructed using a well known formula via Noether's theorem. This has been extended to non variational systems too. This association between symmetries and conservation laws has initiated the double reduction of di erential equations, both ordinary and, more recently, partial. We apply these techniques to a number of well known equations like the damped driven Schr odinger equation and a transformed PT symmetric equation(with Schr odinger like properties), that arise in a number of physical phenomena with a special emphasis on Schr odinger type equations and equations that arise in Optics.

It has been almost eighty years since quantum mechanics emerged as a complete theory, yet debates about how should quantum mechanics be interpreted still occur. Interpretations are many and varied, some taking us as fundamental in determining reality (orthodox interpretation), while others proposing that reality exists outside of us, but it is a lot more complicated than that implied by classical mechanics. In this thesis I am going to try to provide new light on this debate by investigating dynamics under both the orthodox and modal interpretation. In particular I will answer the question what is the interpretation of non-Markovian stochastic Schrodinger equations? I conclude that under the orthodox view these equations have only a numerical interpretation. They provide a rule for calculating the state of the system at time t if we made a measurement on the bath (a collection of oscillators {ak}) at that time, yielding results {zk}. However in the modal view they have a meaning: non-Markovian stochastic Schrodinger equations represent the evolution of the system part of the property state of the universe (bath + system).

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The cubic focusing nonlinear Schrodinger equation is a canonical model equation that arises in physics and engineering, particularly in nonlinear optics and plasma physics. Cubic NLS is an accessible venue to refine techniques for more general nonlinear partial differential equations. In this thesis, it is shown there exist solutions to the focusing cubic nonlinear Schrodinger equation in three dimensions that blowup on a circle, in the sense of L2-norm concentration on a ring, bounded H1-norm outside any surrounding toroid, and growth of the global H1-norm with the log-log rate. Analogous behaviour occurs in higher dimensions. That is, there exists data for which the corresponding evolution by the cubic nonlinear Schrodinger equation explodes on a set of co-dimension two. To simplify the exposition, the proof is presented in dimension three, with remarks to indicate the adaptations in higher dimension.

The cubic focusing nonlinear Schrodinger equation is a canonical model equation that arises in physics and engineering, particularly in nonlinear optics and plasma physics. Cubic NLS is an accessible venue to refine techniques for more general nonlinear partial differential equations. In this thesis, it is shown there exist solutions to the focusing cubic nonlinear Schrodinger equation in three dimensions that blowup on a circle, in the sense of L2-norm concentration on a ring, bounded H1-norm outside any surrounding toroid, and growth of the global H1-norm with the log-log rate. Analogous behaviour occurs in higher dimensions. That is, there exists data for which the corresponding evolution by the cubic nonlinear Schrodinger equation explodes on a set of co-dimension two. To simplify the exposition, the proof is presented in dimension three, with remarks to indicate the adaptations in higher dimension.

The small dispersion limit of the focusing nonlinear Schroödinger equation (fNLS) exhibits a rich structure with rapid oscillations at microscopic scales. Due to the non self-adjoint scattering problem associated to fNLS, very few rigorous results exist in the semiclassical limit. The asymptotics for reectionless WKB-like initial data was worked out in [KMM03] and for the family q(x, 0) = sech^(1+(i/∈)μ in [TVZ04]. In both studies the authors observed sharp breaking curves in the space-time separating regions with disparate asymptotic behaviors. In this paper we consider another exactly solvable family of initial data, specifically the family of centered square pulses, q(x; 0) = qx[-L,L] for real amplitudes q. Using Riemann- Hilbert techniques we obtain rigorous pointwise asymptotics for the semiclassical limit of fNLS globally in space and up to an O(1) maximal time. In particular, we find breaking curves emerging in accord with the previous studies. Finally, we show that the discontinuities in our initial data regularize by the immediate generation of genus one oscillations emitted into the support of the initial data. This is the first case in which the genus structure of the semiclassical asymptotics for fNLS have been calculated for non-analytic initial data.

Dissertação (mestrado)—Universidade de Brasília, Instituto de Física, 2010. / Submitted by Larissa Ferreira dos Angelos (ferreirangelos@gmail.com) on 2011-05-12T18:44:24Z No. of bitstreams: 1 2010_DeiseTamaradosSantosCavalcanteMachado.pdf: 559529 bytes, checksum: 41f72ac9507381f8e1ecb5ae03ee1cba (MD5) / Approved for entry into archive by Daniel Ribeiro(daniel@bce.unb.br) on 2011-05-21T00:00:48Z (GMT) No. of bitstreams: 1 2010_DeiseTamaradosSantosCavalcanteMachado.pdf: 559529 bytes, checksum: 41f72ac9507381f8e1ecb5ae03ee1cba (MD5) / Made available in DSpace on 2011-05-21T00:00:48Z (GMT). No. of bitstreams: 1 2010_DeiseTamaradosSantosCavalcanteMachado.pdf: 559529 bytes, checksum: 41f72ac9507381f8e1ecb5ae03ee1cba (MD5) / Este trabalho se baseia em resolver a equação do tipo Schrödinger, obtida usando a equação de Hamilton-Jacobi (HJE), separada em coordenadas elípticas, sob o método desenvolvido por Hylleraas associado às séries de Wind-Jaffé, onde utilizamos um algoritmo computacional, em Maple, para calcular as soluções dos determinantes das matrizes obtidas por meio de recorrências das séries, e assim obter os níveis exatos de energias eletrônicas para os estados fundamentais e excitados dos íons heteromoleculares HeH2+ , LiH3+ e BeH4+ em função da distância nuclear, além da posição de equilíbrio e a energia de dissociação através da curva de energia eletrônica. _________________________________________________________________________________ ABSTRACT / This work is based on solving the Schrödinger equation type obtained using the Hamilton-Jacobi equation (HJE) in elliptical coordinates, trough the method developed by Hylleraas and the series obtained by Wind-Jaffé. We developed an algorithm for computing the determinants of the matrices obtained through a recurrence series, and obtained the exact values of the electronic energy levels for the ground and excited states of hetero-molecular ions HeH2+ , LiH3+ and BeH4+ as nuclear distance function, and furthermore the equilibrium position and energy of dissociation via electronic energy curve.

Magíster en Ciencias, Mención Física / Los solitones son el fen omeno universal m as profundamente estudiado, debido a los innumerables sistemas físicos en los cuales se observa. Estas soluciones corresponden a estados localizados y coherentes que surgen naturalmente en sistemas extendidos, siendo una de sus propiedades m as fascinantes el hecho de que pueden ser tratados como partículas macroscópicas a pesar de estar formados por numerosos componentes microscópicos. Desde su primera descripci on, realizada por J. S. Russell en 1884, el estudio de solitones se centró en sistemas conservativos por más de cien años. Sin embargo, los pioneros trabajos de Alan Turing e Ilya Prigogine demostraron que los sistemas fuera del equilibrio se auto{ organizan por medio de la generación de estructuras disipativas. Hoy en día, sabemos que es justamente este mecanismo el que permite la formación de solitones disipativos en sistemas con inyección y disipación de energía. Nuestro principal interés ha sido caracterizar de forma analítica y numérica a los solitones que emergen en sistemas forzados paramétricamente{sistemas forzados por medio de un parámetro efectivo que var a en el espacio y/o tiempo. Los sistemas forzados param etricamente pueden experimentar una resonancia paramétrica, la cual se caracteriza por una respuesta subarm onica (subm ultiplos de la frecuencia natural del sistema). Dada la complejidad que presentan los sistemas paramétricos, focalizamos nuestro estudio en la ecuación de Schrödinger no lineal disipativa forzada paramétricamente (PDNLS). Este modelo caracteriza bien la din amica de sistemas forzados param etricamente, en torno al punto de aparición de la resonancia paramétrica, en el límite de baja disipación e inyección de energía. Los solitones disipativos, presentes en PDNLS, típicamente muestran una estructura de fase uniforme. Dichas estructuras han sido ampliamente utilizadas para describir a los solitones hidrodinámicos que aparecen en el experimento de Faraday, estados localizados de la magnetización en un hilo magnético, o los clásicos solitones presentes en una cadena de péndulos con soporte verticalmente vibrado, entre otros. Por medio de simulaciones numéricas interactivas de solitones disipativos en la ecuaciónPDNLS, hemos logrado observar una interesante din amica de frentes de fase hasta ahora desconocida. Estos frentes de fase se propagan hasta alcanzar un punto de equilibrio estacionarioarbitrario. A este tipo de solitones los hemos llamado solitones escudados por la fase (phase shielding solitons), dado que la estructura nal de fase pareciera proteger al módulodel solit on. Hemos logrado caracterizar anal ticamente estas soluciones localizadas, determinando ocho posibles con guraciones. Los solitones estudiados poseen una talla característica dada por el tamaño de la estructura de fase estacionaria. Adem ás, extendimos nuestro estudio al caso bidimensional, mostrando los resultados, dos tipos de phase shilding solitons bidimensionales; axialmente simétricos y asimétricos. Los primeros pueden ser entendidos como una rotación en 2 de las soluciones simétricas encontradas en el caso unidimensional. Por su parte, las soluciones asimétricas bidimensionales presentan propiedades mucho más interesantes, ya que su estructura nal de fáse contiene todas las con guraciones halladas en el caso unidimensional. Con el n de corroborar la existencia de solitones disipativos con estructura de fase no uniforme en sistemas físicos, realizamos simulaciones numéricas de diversos sistemas paramétricos reales. Satisfactoriamente, concluimos que el fenómeno phase shielding soliton es universal, y esperamos que pueda ser prontamente observado experimentalmente.